Searches , social queries for HYPERGEOMETRIC FUNCTION
Search references for HYPERGEOMETRIC FUNCTION. Phrases containing HYPERGEOMETRIC FUNCTION
See searches and references containing HYPERGEOMETRIC FUNCTION!
Search references for HYPERGEOMETRIC FUNCTION. Phrases containing HYPERGEOMETRIC FUNCTION
See searches and references containing HYPERGEOMETRIC FUNCTION!HYPERGEOMETRIC FUNCTION
Function defined by a hypergeometric series
ordinary hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes many other special functions as
Hypergeometric_function
Family of power series in mathematics
generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n {\displaystyle n} is a rational function of n
Generalized hypergeometric function
Generalized_hypergeometric_function
Solution of a confluent hypergeometric equation
a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential
Confluent hypergeometric function
Confluent_hypergeometric_function
Q-analog of hypergeometric series
by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the
Basic_hypergeometric_series
Hypergeometric function in mathematics
mathematics, a general hypergeometric function or Aomoto–Gelfand hypergeometric function is a generalization of the hypergeometric function that was introduced
General hypergeometric function
General_hypergeometric_function
Sigmoid shape special function
Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function (Kummer's function): erf ( x ) = 2 x π M ( 1 2 , 3 2 , − x 2 ) . {\displaystyle
Error_function
Elliptic analog of hypergeometric series
is an elliptic function of n {\displaystyle n} , analogous to generalized hypergeometric series where the ratio is a rational function of n {\displaystyle
Elliptic hypergeometric series
Elliptic_hypergeometric_series
Extension of the factorial function
functions can be expressed in terms of the gamma function. More functions yet, including the hypergeometric function and special cases thereof, can be represented
Gamma_function
Mathematical function
{s^{b-1}}{(1+s)^{a+b}}}\,ds.} It can also be written in terms of the hypergeometric function: B ( x ; a , b ) = x a a 2 F 1 ( a , 1 − b ; a + 1 ; x ) {\displaystyle
Beta_function
mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an
Hypergeometric function of a matrix argument
Hypergeometric_function_of_a_matrix_argument
Generalization of the hypergeometric function
of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's G-function was able to include those as
Meijer_G-function
Topics referred to by the same term
Hypergeometric may refer to several distinct concepts within mathematics: The hypergeometric function, a solution to the Gaussian hypergeometric differential
Hypergeometric
Multivalued function in mathematics
generalization resembles the hypergeometric function and the Meijer G function but it belongs to a different class of functions. When r1 = r2, both sides
Lambert_W_function
Discrete probability distribution
random variable X {\displaystyle X} follows the hypergeometric distribution if its probability mass function (pmf) is given by p X ( k ) = Pr ( X = k ) =
Hypergeometric_distribution
Family of solutions to related differential equations
}e^{-x\sinh t-\alpha t}\,dt.} The Bessel functions can be expressed in terms of the generalized hypergeometric series as J α ( x ) = ( x 2 ) α Γ ( α +
Bessel_function
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Generalization of the Meijer G-function and the Fox–Wright function
Sena Monteiro. "On the Relation between Lambert W-Function and Generalized Hypergeometric Functions". Researchgate. Retrieved 1 March 2023. Mathai, A
Fox_H-function
Types of special mathematical functions
{z^{s+k}}{s+k}}={\frac {z^{s}}{s}}M(s,s+1,-z),} where M is Kummer's confluent hypergeometric function. When the real part of z is positive, γ ( s , z ) = s − 1 z s e
Incomplete_gamma_function
Mathematical theorem on convolved binomial coefficients
case of Gauss's hypergeometric theorem, which states that 2F1(a,b;c;1)=Γ(c)Γ(c−a−b)Γ(c−a)Γ(c−b) where 2F1 is the hypergeometric function and Γ(n+1)=n! is
Vandermonde's_identity
In physics, solution to Schrödinger equation
potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument. The Coulomb wave equation for
Coulomb_wave_function
Polynomial sequence
Gustav Jacob Jacobi. The Jacobi polynomials are defined via the hypergeometric function as follows: P n ( α , β ) ( z ) = ( α + 1 ) n n ! 2 F 1 ( − n
Jacobi_polynomials
Anger–Weber function Kazuhiko Aomoto: Aomoto–Gel'fand hypergeometric function - Aomoto integral Paul Émile Appell (1855–1930): Appell hypergeometric series
List of eponyms of special functions
List_of_eponyms_of_special_functions
Sequence of differential equation solutions
generating function for them is ∑n=0∞tnLn(α)(x)=1(1−t)α+1e−tx/(1−t). Laguerre functions are defined by confluent hypergeometric functions and Kummer's
Laguerre_polynomials
Generalisation of the generalised hypergeometric function pFq(z)
function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric
Fox–Wright_function
Special function in the physical sciences
mathematics, the Airy function (or Airy function of the first kind) A i ( x ) {\displaystyle \mathbf {Ai({\boldsymbol {x}})} } is a special function named after
Airy_function
Polynomial sequence
hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions
Hermite_polynomials
In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman
Bateman_function
Mathematical functions
are increasingly popular. In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and
Falling_and_rising_factorials
Probability distribution
the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n
Binomial_distribution
In mathematics, a solution to a modified form of the confluent hypergeometric equation
mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by
Whittaker_function
Set of four hypergeometric series
of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable
Appell_series
function Riesz function Hypergeometric functions: Versatile family of power series. Confluent hypergeometric function Associated Legendre functions Meijer G-function
List of mathematical functions
List_of_mathematical_functions
Type of mathematical series
two consecutive terms is a rational function of n {\displaystyle n} . The definition of the generalized hypergeometric series is similar, except that the
Bilateral hypergeometric series
Bilateral_hypergeometric_series
Polynomial sequence
Chebyshev polynomials of the second kind. They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite: C n ( α
Gegenbauer_polynomials
Special function defined by an integral
connexion with the confluent hypergeometric functions is that E 1 {\displaystyle E_{1}} is an exponential times the function U ( 1 , 1 , z ) {\displaystyle
Exponential_integral
Contour integral involving a product of gamma functions
product of gamma functions. They were introduced by Ernest William Barnes (1908, 1910). They are closely related to generalized hypergeometric series. The
Barnes_integral
Special function defined by an integral
{i^{k}}{(m+nk+1)}}{\frac {x^{m+nk+1}}{k!}}} is a confluent hypergeometric function and also an incomplete gamma function ∫ x m e i x n d x = x m + 1 m + 1 1 F 1 ( m
Fresnel_integral
Solutions of Legendre's differential equation
which can both be expressed in terms of the hypergeometric function, 2F1. With Γ being the gamma function, the first solution is Pλμ(z)=1Γ(1−μ)[z+1z−1]μ/22F1(−λ
Legendre_function
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
Well defined hypergeometric series discovered by Giuseppe Lauricella
In 1893 Giuseppe Lauricella defined and studied four hypergeometric series FA, FB, FC, FD of three variables. They are (Lauricella 1893): F A ( 3 ) ( a
Lauricella hypergeometric series
Lauricella_hypergeometric_series
Input to a mathematical function
hypergeometric function is an example of a four-argument function. The number of arguments that a function takes is called the arity of the function.
Argument_of_a_function
Number of subsets of a given size
\alpha } . Binomial transform Delannoy number Eulerian number Hypergeometric function List of factorial and binomial topics Macaulay representation of
Binomial_coefficient
Family of orthogonal polynomials
polynomials. Hahn polynomials are defined in terms of generalized hypergeometric functions by Qn(x;α,β,N)=3F2(−n,−x,n+α+β+1;α+1,−N+1;1) for n=0,1,…,N. If
Hahn_polynomials
Mathematical function having a characteristic S-shaped curve or sigmoid curve
functions M25: Special cases of Gauss hypergeometric functions M26: Feedback closed-loop systems M27: Recursive functions M28: Recursive time-delayed feed-forward
Sigmoid_function
Japanese mathematician
is a Japanese mathematician who introduced the Aomoto-Gel'fand hypergeometric function and the Aomoto integral. He was a professor at Nagoya University
Kazuhiko_Aomoto
Pair of functions in combinatorics
involving binomial coefficients, factorials, and in general any hypergeometric series. A function's WZ counterpart may be used to find an equivalent and much
Wilf–Zeilberger_pair
Pair of polynomial sequences
yieldsTn(x)=∑j=0⌊n/2⌋(n2j)(x2−1)jxn−2j.This can be written as a 2F1 hypergeometric function: Tn(x)=∑k=0⌊n/2⌋(n2k)(x2−1)kxn−2k=xn∑k=0⌊n/2⌋(n2k)(1−x−2)k=12n
Chebyshev_polynomials
Analytic function that does not satisfy a polynomial equation
generalized hypergeometric and Bessel functions are transcendental in general, but algebraic for some special parameter values. Transcendental functions cannot
Transcendental_function
of hypergeometric identities. Hypergeometric function lists identities for the Gaussian hypergeometric function Generalized hypergeometric function lists
List of hypergeometric identities
List_of_hypergeometric_identities
Irreducible representation of the rotation group SO
) s i m − m ′ , {\displaystyle (-1)^{s}i^{m-m'},} causing half of the functions to be purely imaginary. The realness of the d-matrix elements is one of
Wigner_D-matrix
–2t/(1–t2) An explicit expression for them in terms of the generalized hypergeometric function 3F0: s n ( x ) = ( − x / 2 ) n 3 F 0 ( − n , 1 − n 2 , 1 − n 2
Mott_polynomials
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
generalized hypergeometric type, and in fact the Bessel–Clifford function is up to a scaling factor a Pochhammer–Barnes hypergeometric function; we have
Bessel–Clifford_function
Special function defined by an integral
In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number
Logarithmic_integral_function
System of complete and orthogonal polynomials
polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials, and associated Legendre functions. In this approach,
Legendre_polynomials
Canonical solutions of the general Legendre equation
gamma function and 2F1 is the hypergeometric function 2F1(α,β;γ;z)=Γ(γ)Γ(α)Γ(β)∑n=0∞Γ(n+α)Γ(n+β)Γ(n+γ) n!zn, They are called the Legendre functions when
Associated Legendre polynomials
Associated_Legendre_polynomials
Special mathematical functions defined on the surface of a sphere
group is given by the hypergeometric series; furthermore, the spherical harmonics can be re-expressed in terms of the hypergeometric series, as SO(3) = PSU(2)
Spherical_harmonics
Concept in mathematics
) {\displaystyle \;_{1}F_{1}(a;b;z)=M(a;b;z)} is the confluent hypergeometric function. Other pairs of independent solutions may be formed from linear
Parabolic_cylinder_function
Type of functions, in mathematical analysis
the class of hypergeometric functions. Examples of special functions that are holonomic but not hypergeometric include the Heun functions. Examples of
Holonomic_function
Mathematical identities related to integer partitions
the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered
Rogers–Ramanujan_identities
Summation method for hypergeometric terms
where S(n) is a hypergeometric term (i.e., S(n + 1)/S(n) is a rational function of n); then necessarily a(n) is itself a hypergeometric term, and given
Gosper's_algorithm
Generalization of the hypergeometric differential equation
equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur
Riemann's differential equation
Riemann's_differential_equation
Probability distribution
particular instance of the hypergeometric function. For information on its inverse cumulative distribution function, see quantile function § Student's t-distribution
Student's_t-distribution
German polymath and scholar (1777–1855)
general hypergeometric function F(α,β,γ,x), and shows that many of the functions known at the time are special cases of the hypergeometric function. This
Carl_Friedrich_Gauss
Special function defined by an integral
the sinc function, and also the zeroth spherical Bessel function. Since sinc {\displaystyle \operatorname {sinc} } is an even entire function (holomorphic
Trigonometric_integral
Discrete probability distribution
special case where α and β are integers is also known as the negative hypergeometric distribution. The beta distribution is a conjugate distribution of the
Beta-binomial_distribution
Special function in mathematics
In mathematics, the Kampé de Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de
Kampé_de_Fériet_function
In mathematics, the E-function was introduced by Thomas Murray MacRobert (1937–1938) to extend the generalized hypergeometric series pFq(·) to the case
MacRobert_E_function
Special functions used to build correlation functions in 2D CFTs
the Virasoro algebra; four-point blocks on the sphere reduce to hypergeometric functions in special cases, but are in general much more complicated. In
Virasoro_conformal_block
Probability distribution
characteristic function of the beta distribution to a Bessel function, since in the special case α + β = 2α the confluent hypergeometric function (of the first
Beta_distribution
Topics referred to by the same term
G-function, related to the Gamma function Meijer G-function, a generalization of the hypergeometric function Siegel G-function, a class of functions in
G-function
Type of function in mathematics
special functions are analytic on a suitable domain: hypergeometric functions on suitable domains Bessel functions on suitable domains The gamma function away
Analytic_function
Measure of linear correlation
is the gamma function and 2 F 1 ( a , b ; c ; z ) {\displaystyle {}_{2}\mathrm {F} _{1}(a,b;c;z)} is the Gaussian hypergeometric function. In the special
Pearson correlation coefficient
Pearson_correlation_coefficient
Triangular array of the binomial coefficients
convolving the distribution function for a random variable with itself corresponds to calculating the distribution function for a sum of n independent
Pascal's_triangle
Orthogonal polynomials
Carl Charlier in 1905. They are given in terms of the generalized hypergeometric function by C n ( x ; μ ) = 2 F 0 ( − n , − x ; − ; − 1 / μ ) = ( − 1 )
Charlier_polynomials
Risk measure estimating the average loss in the worst tail of the distribution
beta function is defined only for positive arguments, for a more generic case the expected shortfall can be expressed with the hypergeometric function: ES
Expected_shortfall
Discrete probability distribution
In probability theory and statistics, the negative hypergeometric distribution describes probabilities for when sampling from a finite population without
Negative hypergeometric distribution
Negative_hypergeometric_distribution
Real root of the polynomial x^5+x+a
series for Bring radicals, as well as a representation in terms of hypergeometric functions can be derived as follows. The equation x 5 + x + a = 0 {\displaystyle
Bring_radical
Recurrence relations of binomial coefficients in Pascal's triangle
distribution · Hypergeometric distribution · Poisson distribution · Gamma distribution Special functions (connections) Gamma function · Beta function · Incomplete
Hockey-stick_identity
Probability distribution
Distribution". Wroughton, Jacqueline. "Distinguishing Between Binomial, Hypergeometric and Negative Binomial Distributions" (PDF). Hilbe, Joseph M. (2011)
Negative binomial distribution
Negative_binomial_distribution
functions are given in terms of the q-Pochhammer symbol and the basic hypergeometric function ϕ {\displaystyle \phi } by J ν ( 1 ) ( x ; q ) = ( q ν + 1 ; q
Jackson_q-Bessel_function
Product of consecutive factorial numbers
continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function as s f ( n ) = G ( n + 2 ) {\displaystyle
Superfactorial
Algebraic expansion of powers of a binomial
{n}{k}}={\frac {n!}{k!\;(n-k)!}},} which is defined in terms of the factorial function n!. Equivalently, this formula can be written ( n k ) = n ( n − 1 ) ⋯ (
Binomial_theorem
In mathematics, Ferrers functions are certain special functions defined in terms of hypergeometric functions. They are named after Norman Macleod Ferrers
Ferrers_function
Type of permutation of a set of elements
Dn=[n!e]=⌊n!e+12⌋for n≥1, where [x] is the nearest integer function and ⌊x⌋ is the floor function. Other related formulas include Dn=⌊n!+1e⌋for n≥1,Dn=⌊(e+e−1)n
Derangement
q\right]} where φ is a basic hypergeometric function, x = cos θ, and (,,,)n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral
Askey–Wilson_polynomials
Measure of internal forces in an atomic nucleus
}{\frac {(a)_{n}(b)_{n}}{(c)_{n}}}{\frac {z^{n}}{n!}}} is the hypergeometric function. It is also possible to analytically solve the eigenvalue problem
Woods–Saxon_potential
Classification of orthogonal polynomials
scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials
Askey_scheme
Mathematical equation
the form of the hypergeometric differential equation. It has two linearly independent solutions, called the periods of elliptic functions. The ratio of
Picard–Fuchs_equation
Linear recurrence equation
quotient of two polynomials. A hypergeometric function f {\textstyle f} is a function in the broad class of analytc functions that can be represented as a
P-recursive_equation
Monochrome light beam whose amplitude envelope is a Gaussian function
real-valued, Γ(x) is the gamma function and 1F1(a, b; x) is a confluent hypergeometric function. Some subfamilies of hypergeometric-Gaussian (HyGG) modes can
Gaussian_beam
Equalities involving sums over the coefficients occurring in hypergeometric series
mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These
Hypergeometric_identity
their properties. The polynomials are given in terms of basic hypergeometric functions by H n ( x | q ) = e i n θ 2 ϕ 0 [ q − n , 0 − ; q , q n e − 2
Continuous q-Hermite polynomials
Continuous_q-Hermite_polynomials
special functions, Schwarz's list or the Schwarz table is the list of 15 cases found by Hermann Schwarz (1873, p. 323) when hypergeometric functions can be
Schwarz's_list
Special function defined by an integral
where n!! denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed
Elliptic_integral
Russian mathematician (born 1962)
investigated generalized Euler integrals, A {\displaystyle A} -hypergeometric functions, A {\displaystyle A} -discriminants, and hyperdeterminants, and
Mikhail_Kapranov
Cunningham (1908). It can be defined in terms of the confluent hypergeometric function U, by ω m , n ( x ) = e − x + π i ( m / 2 − n ) Γ ( 1 + n − m /
Cunningham_function
properties. The q-Laguerre polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by Ln(α)(x;q)=(qα+1;q)n(q;q)n1ϕ1(q−n;qα+1;q
Q-Laguerre_polynomials
Transformation of a mathematical sequence
R. B. (2010). "Euler-type transformations for the generalized hypergeometric function". Z. Angew. Math. Phys. 62 (1): 31–45. doi:10.1007/s00033-010-0085-0
Binomial_transform
theory of special functions in mathematics, the Horn functions (named for Jakob Horn) are the 34 distinct convergent hypergeometric series of order two
Horn_function
travel, tourism, insurance
HYPERGEOMETRIC FUNCTION
HYPERGEOMETRIC FUNCTION
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Biblical
Look for pages within Wikipedia that link to this title
If a page was recently created here it may not be visible yet because of a delay in updating the database; wait a few minutes or try the function.
Look for pages within Wikipedia that link to this title
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Male
Egyptian
, a high Egyptian functionary.
Male
Egyptian
, a great functionary.
Male
Egyptian
, an Egyptian functionary.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Celtic
, great justiciary, or functionary.
Male
Egyptian
, an Egyptian functionary.
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Male
Egyptian
, the son of the functionary Heknofre.
Male
Egyptian
, Functionary of the Interior.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
HYPERGEOMETRIC FUNCTION
HYPERGEOMETRIC FUNCTION
HYPERGEOMETRIC FUNCTION
HYPERGEOMETRIC FUNCTION
HYPERGEOMETRIC FUNCTION
HYPERGEOMETRIC FUNCTION
HYPERGEOMETRIC FUNCTION
travel, tourism, insurance